I want to program an application on iPhone and I want to know if the idea of this application is doable.
So in the Application I want to have an image that appears to the users (this is easy. When the user touches the image, the application records the (x,y) position of where the user touches exactly in the image and stores it in a file (.pdf, .txt or what ever)
I need these (x,y) positions to be stored in some kind of file but I don't know if this is doable in iPone or iPad

sure. Each app gets its own place to save things to. You would write out a file just like any other program. Some locations are meant to be temporary, some are meant to be backed up when synced with iTunes.